The underlying manifold of a low-volume hyperbolic 3-orbifold

Abstract

This paper proves strong restrictions on the underlying topological space of a closed, orientable hyperbolic 3-orbifold \orbM whose volume satisfies a certain upper bound. For instance, if vol(\orbM) < 0.1491, then the underlying space of \orbM must be either a small Seifert fibered space, or the connected sum of two lens spaces (or of a lens space with S^2 x S^1), or the gluing of one or two highly restricted Seifert fibered spaces along a single incompressible torus. If vol(\orbM) < 0.1571 and the singular locus of \orbM is a link, then the topology of the underlying space is restricted even further. Our methods are primarily topological, and involve studying the underlying topology of orbifold books of I-bundles. Along the way, we provide an exposition of some foundational material about orbifolds that was previously absent from the literature.

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BibTeX RIS

BibTeX

@misc{futer2026underlying,
  title = {{The underlying manifold of a low-volume hyperbolic 3-orbifold}},
  author = {David Futer and Peter B. Shalen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16212},
  eprint = {2609.16212},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16212},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The underlying manifold of a low-volume hyperbolic 3-orbifold
AU  - David Futer
AU  - Peter B. Shalen
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.16212
UR  - https://arxiv.org/abs/2609.16212
AB  - This paper proves strong restrictions on the underlying topological space of a closed, orientable hyperbolic 3-orbifold \orbM whose volume satisfies a certain upper bound. For instance, if vol(\orbM) < 0.1491, then the underlying space of \orbM must be either a small Seifert fibered space, or the connected sum of two lens spaces (or of a lens space with S^2 x S^1), or the gluing of one or two highly restricted Seifert fibered spaces along a single incompressible torus. If vol(\orbM) < 0.1571 and the singular locus of \orbM is a link, then the topology of the underlying space is restricted even further. Our methods are primarily topological, and involve studying the underlying topology of orbifold books of I-bundles. Along the way, we provide an exposition of some foundational material about orbifolds that was previously absent from the literature.
ER  -

Abstract

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